Let $T : \mathcal A \rightarrow \mathcal B$ be a functor. Let $\mathcal C$ be the category whose objects are the objects of $\mathcal A$ and whose morphisms are given as $\text{Hom}_{\mathcal C}(A,B) = \text{Hom}_{\mathcal B}(TA,TB)$. Composition is defined using composition in $\mathcal B$.
Is there a neater way to construct the category $\mathcal C$? I'm thinking maybe of using comma categories or other more general notions but can't actually come up with a category isomorphic to $\mathcal C$ using these methods.
Thanks in advance for the help.