Natural maps between sets of linear maps Let $\mathcal{L}(V,W)$ denote the set of linear maps $T: V \to W$. For finite-dimensional vectors spaces $U,V,W$, I am trying to find a natural map $\mathcal{L}(V,W) \to \mathcal{L}(V,U)$ and one from $\mathcal{L}(U,V) \to \mathcal{L}(W,V)$. I'm struggling to get started due to how abstract the notion of a 'map of maps' is. In particular, I can't do anything of the form $Tv \mapsto Tw$ or even any kind of inverse or composition because the dimensions don't line up.
Any guidance would be appreciated.
Given a fixed linear transformation $\varphi : W \rightarrow U$, you can define the transformation $T : \mathcal{L}(V,W) \rightarrow \mathcal{L}(V,U) $ defined for all $u \in \mathcal{L}(V,W)$ by
$$T(u) =\varphi \circ u$$
I guess this is the most natural thing you can do !
Because your vector spaces are all finite-dimensional, the linear maps can be realized as matrices over the underlying field of the vector spaces (which are assumed fixed over all $U,V,W$. Now note that we have the following canonical (respectively, left- and right-) group actions via multiplication by of $\mathcal{L}(W,U)$,
$$\alpha\colon \mathcal{L}(W,U) \times \mathcal{L}(U,V)\to \mathcal{L}(W,V)\,,~\,~\,~(M,N)\mapsto MN$$
$$\beta\colon \mathcal{L}(V,W)\times \mathcal{L}(W,U)\to \mathcal{L}(V,U)\,,~\,~\,~(N,M)\mapsto NM\,.$$
You can then restrict to a fixed element of $\mathcal{L} (W,U)$ to obtain the desired natural maps.