with my hook fast in a corner of his mouth. (lines 3-4)
The beginning of the poem demonstrates the battle as we might expect it: man being the aggressor in the natural world. Here, we see by the pronouns Bishop uses that it's "my" hook in "his" mouth, instead of a much more hands-off choice of pronouns like, say, "a" hook in "its" mouth.
I looked into his eyes which were far larger than mine but shallower (lines 34-36)
In line 34 you think we might be seeing the beginning of a connection or a "look we're all the same in the end" message, but lines 35 and 36 go on to correct that. It's still very much a man vs. nature scenario.
They shifted a little, but not to return my stare. (lines 41-42)
Much like lines 34-36 this shows how the human and the fish are <em>not</em> alike. It's here, too, that we start to see that although this is a man vs. nature situation, the real conflict lies very much within the human.
still crimped from the strain and snap when it broke and he got away. (lines 58-59)
OK, so this is not evidence of their man vs. nature battle, but of previous such battles. And because seeing the fishing lines affect the speaker so much, we think it's important to point out here.