Degrees of freedom for injective endofunctor keeping objects constant?

Question

Suppose we have a category $C$ with at least 2 objects. In fact, I have in mind the category Set, but my question applies to all categories.

Suppose we define a functor $F:C\to C$ that

1. Maps objects to themselves: $F_o(X)=X$

2. Is injective in endomorphisms: for all $f,g:X\to X$ we need $F_m(f)=F_m(g)\implies f=g$.

How many degrees of freedom do we have for defining such a functor?

My current conjecture is: Our space of such functors is exactly categorized by:

1. For each object $X$, there has to be a monoid isomorphism between $\text {Hom}(X,X)$ and $F[\text{Hom}(X,X)]$.

2. For each object $X$, we are allowed to choose exactly one isomorphism $\pi_X:X\to X$, so that for all objects $Y\neq X$ and morphisms $f:X\to Y$ and $g:Y\to X$ we have $F(f)=\pi_Y^{-1}\circ f\circ \pi_X$, and $F(g)=\pi_X^{-1}\circ g\circ \pi_Y$.

Is this correct?

I also want to think about how the degrees of freedom change if we relax the assumption that objects are mapped to themselves, but keep the assumption that the functor is injective.