On the website nlab looking for examples for essentially surjective functors I found the following affirmartion: any bijective on objects functor is essentially surjective. I can use surjectivity of the functor that leads me to an equality that allows me to get the isomorphism asked on the essential surjective, this way: Let $F:C\rightarrow D$ be a bijective on objects functor, so for $d\in D$, exists $c\in C$ such that $F(c)=d$, so we can take the identity functor, which is an isomorphism to obtain $F(c)\simeq d$. My problem is that I can not figure out where to use the injectivity, I am missing something?
Thank you
You're not missing anything. The conclusion holds assuming the functor is just surjective on objects. Indeed, the entire point of "essentially surjective" is that it is a mild weakening of "surjective on objects" (hence the name).