I would like to post my answer as well. In the Chapter 3 of my thesis:
"Automorphism-preserving color substitutions on Profinite Graphs" (2022). Electronic Thesis and Dissertation Repository. 8795.
https://ir.lib.uwo.ca/etd/8795 ,
I came up with a certain categorical notion that worked for what I intended to do: prove the common properties of structures such as profinite groups, graphs and rings for one general category. The name: profinite category is already being used for a different approach: see for example the paper by J. Almeida and P. Weil https://www.sciencedirect.com/science/article/pii/S0022404996000837 , or the post by @Benjamin Steinberg. I called my category instead a Profinite structure.

Since profinite objects are essentially completions of finite objects, I decided that I first needed a functor $F$, into the category of sets that let me make a difference between finite and infinite objects. I also wanted these completions to exist, so I required that $F$ transforms a projective limit of finite objects into a projective limit of sets. With these points in mind I defined a "preprofinite" category, which is essentially the category of objects I want to complete: one can think of for example rings or groups. The definition of a preprofinite category is:

A couple $(\mathcal{C},F)$ with $\mathcal{C}$ being a category and $F$ a functor with:

- $F$ faithful
- If $(X_i)_{i \in I}$ together with transition maps $\phi_{i,j}$ is a a projective system in $\mathcal{C}$, such that $F(X_i)$ is finite for all indexes $i \in I$, then there the system has a projective limit. Furthermore if $(X,(\pi_i)_{i \in I})$ is a projective limit of such a system, then $(F(X),(F(\pi_i)_{i \in I})$ is a projective limit of the projective system $(F(X_i))_{i \in I}$ with $F(\phi_{i,j})$ being the transition maps.
- For all objects $A,B,C$ in $\mathcal{C}$ and for all morphisms $u$ from $B$ to $A$, $v$ from $B$ to $C$ and $f$ a map from $F(A)$ to $F(C)$, if $F(u)$ is surjective and $f \circ F(u)=F(v)$, then there exists a morphism $w: A \rightarrow C$, such that $f=F(w)$.

Typically one should think of $F$ as a forgetful functor. The last axiom establishes a certain relation between maps and maps that are morphisms and was useful to me to prove the results that I needed.

Now that we defined what I called a preprofinite category, one can simply define a profinite stucture as a completion of finite objects in that category. That is:

If $(\mathcal{C},F)$ is a preprofinite category, define a profinite structure on $(\mathcal{C},F)$ as a category $\mathcal{P}$, whose objects are limits of projective systems of finite objects in $\mathcal{C}$ and whose morphisms are morphisms $u$ in $\mathcal{C}$, such that $F(u)$ is continuous for the induced profinite topologies on the sets.