The question I ask myself about these sorts of things is what exactly is it I would like to accomplish? Well, in this case, many things.
Prevent rust.
Prevent wear (keep moving parts, especially highly-loaded ones, from touching.
Stay put and not migrate.
Do all those things consistently in all service temperatures.
So we start with a base oil because oil film's oxygen barrier prevents air getting to the metal (or below the finish)l to rust it. We want it to be load bearing and have high film strength (which is NOT a natural attribute of liquids) so we add extreme pressure agents (like ZDDP found in STP in only trace quantities nowadays) to deal with boundary lubrication conditions. We want it to stay put so we add some eschtickenzie to it (polybutene, the universal stickifier found in bar oil, Lucas products, bird repellant etc which also adds some film strength and a LOT of viscosity) ,....but wait now it gums up in the cold, builds up with use, and makes varnish. And we forgot to add anything to keep our oil itself from oxidizing because we chose the wrong base oil for the job in the first place. So quick, add some solventy stuff to keep the gum in a slurry (but not too much, you don't want to defeat the eschtickenzie! ) and while we're at it add some pour-point depressants so it works better in the cold.
Oh...but now you want a CLEANER as well as a lubricant and metal protectant? Great, just add some more solventy stuff, but mow you're gonna need some more eschtickenzie to make it stay out and now your viscosity and gum levels are back up. Oh well, just use it more often, right? Keep things cleaned so it doesn't gum up, keep plenty on there so it lubes, and keep the excess wiped off so it doesn't get everywhere you don't want it.
Guess what? If you're gonna actually do the MAINTENANCE, all you need is mineral spirits and plain mineral oil. Some parts always do and always have needed actual GREASE, so use that sparingly as necessary and don't ask your CLP to cover a job it can't.
Mote on the oils. Engine oils are more than 25% additive package by volume and those additives are mostly to make the oil do stuff it needs to at a couple hundred degrees inside an engine, not your firearm. So you think you got a real "synthetic", do you? Prove it. Synthetics that do what you want them to (have low viscosity indeces (VI) without the addition of coiled, long-chain polymers or pour-point depressants) are either of the man-made polyalphaolephin (PAO) or ester/polyolester (POE) variety (there are others like PAG which won't apply to guns). Mobil1 USED to be a PAO but (long story) is not anymore. The only true synthetics I know if are made by Amsoil, Royal Purple, and Redline. What does this mean? Well, your engine oil has a bunch of crap in it you may or may not want on your guns like heavy detergents, dispersants, and over-base pH additives. Mineral oils like Dexron III and laxative-grade white oil don't gum or varnish and have a naturally high VI which does well in both high and low temperatures without drama, excepting extreme cold. Below 10⁰F you'd probably better either run your guns dry or use a TRUE PAO Synthetic oil, if you can find one (there is no truth in oil advertising thanks to the Supreme Court ruling in Mobil vs. Castrol).
Very few true synthetic greases exist outside of the aviation industry. If it doesn't specifically advertise being a PAO or Ester blend base oil, it's just ordinary hydrocracked dinosaur juice and suffers all the downsides of such.
If it is and open system and needs to slide under pressure in all temperatures, the only things that really work are hard coatings like Teflon and ceramics.
Here's what I use for all of my guns:
There's a light and heavy synthetic grease, Ed's Red for bore and chamber care, RP gun oil for lube and wiping down, and Hoppes' Benchrest #9 for ambiance and gently removing copper fouling on the rare occasion I need either.
Not shown is a clean parts vat full of Safety Solvent 140 (Varsol) in the shop for full strip/clean jobs.