The differentiation operator $D$ takes a function $f:A\to B$ between differentiable manifolds $A$ and $B$ and assigns to it a function $Df : A_0 \to \mathcal{L}(A,B)$ which in turn assigns, to each point $a_0\in A$ at which $f$ is differentiable (we'll call the set of these points $A_0$), the total derivative of $f$ at $a_0$.
Now, I tried to "formally" define this operator with something along the lines of
$$ D : \operatorname{Hom}_{\mathbf{Set}}(A, B) \to \operatorname{Hom}_{\mathbf{Set}} (A_0, \operatorname{Hom}_{\mathbb{R}\mathbf{-mod}}(A, B)), $$
but, needless to say, there seems to be much wrong with my attempt (it needs $A$, $B$, and, most tragically, somehow $A_0$, to be known in advance).
How could I go about "formally" writing down the definition I had written above in words in a compact (i.e. without any words), rigorous, category-theoretic (or so) way?
I'd want to do this without having to fix anything in advance, i.e. ideally we wouldn't be defining a scheme to define these operators, but rather just have "one operator to rule them all". Maybe someone with more experience in category theory or functional analysis can help me out, or point me in the right direction. Thank you!