One of mine is easy; I leave the carrier in its normal horizontal position, pick her up, point her in head first and shove. She goes right in. And hates the vet so much, she voluntarily goes back in when she's done.
The other one? My demon
child cat? Whoo boy. The first time, I was wearing leather gloves because she's a biter. And then she got out of the dang thing. What I usually do is take the carriers out of storage about a week before the appointment and leave them out and accessible. Then, the day of the appointment, whenever demon cat is calm and settled and it is reasonably close to the appointment, I just pick her up. Which she hates. The complaining begins immediately. Her carrier, I put up on end and I drop her in butt first then slam the door. Once I have it latched, I return the carrier to its proper position.
I'm just really happy they don't have front claws when I have to do this. Saves my skin from being shredded. (I didn't declaw them; they came that way when I got them from a family that couldn't keep them because their toddler son developed life threatening asthma and the specialist told them to get rid of their dog, cats and birds until he was under control. They could keep the fish. The Mom was heartbroken.)