Identity-On-Objects Functor

Traditionally, an identity-on-objects functor $F:\mathcal{C}\to\mathcal{D}$ is a Functor with $F(X)=X$. This definition clearly violates the Principle of Equivalence, but makes sense if we consider $\mathcal{C}$ and $\mathcal{D}$ to be a different class of algebraic structure; EG, Strict Categories.

Alternatively, we can view the functor $F$ as either a Codisplayed Category, or as fitting into an Short Exact Sequence of categories $\mathcal{C}\xrightarrow{F}\mathcal{D}\xrightarrow{mathrm{Id}{ifx..lse\par i}} \mathcal{D}$.