Our route for Cathedral Peak basically follows that suggested by G&M. The couloir to contend with drops from the first major saddle south of the summit. Below the couloir, the terrain fans out into a large rock glacier that occupies most of the valley below the SE face of Cathedral. When you arrive at the lake, looking at the NW end of the lake, there is a large, prominent patch of green comprised mostly of willows & tundra. Slightly above and to the right is a fainter tundra patch and above there a little more tundra. You will want to utilize those areas as a means of gaining this basin and avoiding as much of the talus as possible.
Once you arrive at Cathedral Lake, follow a trail to the outlet for the lake at its most northern point. Cross the stream and then head up some and west over to the first green patch of willows & tundra. Head up along the east side avoiding the willows and then follow the easiest path you can find into the rubble-filled basin SE of the summit. G&M mention an "old but well-preserved miners trail" leading up into this basin. In our 1993 account, we mention this trail. To find it, we took the trail toward Electric Pass a little bit, then turned west and south a little contouring below the prow of the mighty, rugged and spectacular east ridge of the peak. we intersected the trail at about 12,200 feet just below that prow. That trail led us into the SE basin climbing steeply on tundra then rocks before things leveled out some. In 1993, this trail was already rather vague. Two and a half decades more of rockfall may have all but obliterated it so don't count on it. Some faint sections may still be identified on GE.
In the SE basin below Cathedral, there will appear to be three possible couloirs. identify the proper couloir. It will be the northernmost of the three. We have provided a photo to assist. Head to the base of it on almost all rocky rubble unless it's early season where you will likely find a lot of snow. The snow makes progress much easier. If the basin is filled with snow, the couloir will be as well. This gully is quite steep so when you reach the base, we suggest getting out the ice axe and strapping on the crampons if you have them. Carrie only had "instep" crampons, which we found not very secure because the angle was so steep. Micro-spikes may help but will not be as effective as crampons. Be prepared to spend more time in the couloir than you would expect if you have to kick steps. Some parties or party members may enjoy the security of being roped up. If so, some sling, extra biners and harnesses may be useful. If the couloir is not snow-filled it will be as G&M describe, "a tedious, tilted sand box where poor footing and gravity foil your best efforts." Before we headed up into the couloir, we contemplated going directly up one of the gullies on the SE face of Cathedral. Our climbing companion argued admirably for attempting that way, but G&M offer this advice: "The map contours do not indicate that this face is particularly steep; however, the nature of the rock leaves climbers wishing they had ascended the gully (couloir) instead." So, they prefer the tilted sand box over the SE face gullies.
At the head of the couloir, if snow-filled, you may have a cornice to deal with. We found it melted out enough on the left side to get around without having to get too technical. Once out of the couloir, head north along the ridge to the summit. This final approach may also take longer than expected. It's never difficult, but there are several cliff bands to navigate around & through and you're still hiking on a lot of loose, broken rock similar to so many other Elk peaks. Shortly before arriving at the summit section where things level out, there is a more significant cliff band & something of a notch that may send you to the west to get around. When you arrive at the summit, the view is quite stunning with much of the Elk Range visible and Cathedral Lake glistening below. For the descent, we recommend descending as you came, but going down the couloir, especially on snow may be intimidating because of its steepness. Once back at Cathedral Lake, you will welcome the trail hike back to the trailhead.
It should be noted that from the summit of Cathedral it is possible to follow the rugged north ridge to Electric Pass Peak. The route through all the crags & pinnacles is a minimum 3rd class and possibly 4th class depending on the exact route chosen. Most of the difficulties are avoided by traversing on the west side. A description of this traverse may be found under Cathedral Peak on Lists of John.
Links to other information, routes & trip reports for this peak that may be helpful.