Gifts from other Cats?

I've got a male cat "McChouffe" who is an indoor cat cause he was declawed by his previous owner. McChouffe is allowed out on our second floor deck and has been trained to stay on it. There are stairs leading up to the deck and he could get down if he really wanted to and has in the past. The last time he did get off he did so to fight with his Nemesis (McChouffe got his but kicked and hasn't left the deck since). Nemesis is a neighborhood cat (male) who likes to antaganize him from the windows. Well yesterday morning before McChouffe got to go out on the deck we found a head of a bird on the top steps just outside the gate of the deck and the body of the bird down at the bottom of the stairs. We cleaned it up before he got to see it but he could smell that something happened and jumped up on the rail by the gate pondering leaving the deck to investigate. We caught him in time and took him in, but knew why he did it.
We've never fed nor offered affection to Nemesis and now that we have McChouffe and seen his reactions to Nemesis I clap my hands and chase off Nemesis if I see him around.
What I'm not sure of is, should we have left the head there for him see. Not sure what the message was, was it just a gift for us or a peace offering for him or a in your face antaganizing jester for him. I've had other cats that we've owned back when I was a kid, who used to bring us gifts of dead mice etc and can understand cats bringing gifts to their owners. But leaving one for someone else baffles me. Any experiences with this, what do you think?

Wow! That was very The Godfather of Nemesis! I'd hire a bodyguard for your household. You can't take these threats too seriously!

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Ok, serious reply. Several scenarios come to mind. Is it possible that Nemesis killed the bird and was eating/playing with it when you can outside and he ran off? Second, he desires a romantic fling with your cat?! Third, he's claiming your deck as his territory and leaving his dead prey there is a signal to your cat. I like the implications of the second idea but I think the third is the most likely answer!

Ok, maybe that wasn't terrilbly serious.

Well they are both Males... So I don't think theres any romance.
Nemesis has gotten up on the deck and sprayed the area by the back door I've see signs of this on the siding. But this usually happens in the wee hours of the night when no one is around.

One Thanksgiving we were headed out the door to go to sis's house for dinner. One of our females ran out and there on the walk was a dead rabbit. She grabbed it and took off. We have always wondered if it had been left for her by a neighbor's cat. (This was back in the olden days when we could let cats out.)

What is really funny is that after we arrived at sis's house, her hubby, son,and son-in-law came in from rabbit hunting. They didn't get any!

I think it's a territorial gesture.

Hmmmmm the only thing other cats bring to me is their pee!!!!!!

Not joking there is a female cat a couple of doors down who comes to annoy humphrey and then pee on the utility room bit outside our house! Sometimes wish she would stop playing hard to get!!! LOL

Grrrrr glad we are moving soon!!!!