If we have three $k-$vector spaces $V,W,U$, then, for a linear map $T:V\to W$ we have an induced map $T^*:L(W,U)\to L(V,U)$ defined by $T^*(f)=fT$. This map is usually called the transpose of $T$.
But we also have another induced map $T_*:L(U,V)\to L(U,W)$ defined by $T_*(f)=Tf$. Does this one have a particular name?