I'm working through these notes, which cover some fixed point theory for categories.
Here, we encounter the following definitions (I've rephrased them):
Let $C$ be a category and $F$ an endofunctor on $C$.
A prefixed point of $F$ is pair $(A, \alpha)$ where $A$ is an object of $C$ and $\alpha$ a morphism from $F(A)$ to $A$;
a fixed point of $F$ is a prefixed point $(A, \alpha)$ where $\alpha$ is an isomorphism.
Then. the category PFP of prefixed points of $F$ is defined in the following way:
its objects are the prefixed points of $F$;
for each $(A,\alpha)$, $(B,\beta)$ in PFP, the morphism set from $A$ to $B$ contains exactly the arrows $f \in \hom(A,B)$ such that the diagram $$ \begin{aligned} &F(A) &\xrightarrow{\alpha} &A \\ F(f) &\downarrow & &\downarrow f\\ &F(B) &\xrightarrow{\beta} &B \end{aligned} $$ commutes.
Can I then define the category FP of fixed points of $F$ as the full subcategory of PFP whose objects are the fixed points of $F$?