Let $T:V \to W$ be a linear map between two finite-dimensional vector spaces. The dual map of $T$ is $T^*:W^* \to V^*$. $V^*$ map be identified with $V$ in an unique way, by the mapping $\mathcal I_V:V^* \to V$, such that $$ \langle \mathcal I_V(\phi), v \rangle = \phi(v),\forall v\in V, $$ and similarly we have $\mathcal I_W$ for $W$. We write $\phi^\sharp$ for $\mathcal I_V(\phi)$ and similarly for elements of $W^*$, and this is called musical isomorphism.
Question: what notation should one use for the composition $$ \mathcal I_V\circ T^* \circ\mathcal I_W^{-1} ? $$ I find it difficult because I am raising an index of $T^*$ and lowering the other index, so I cannot use a simple $\sharp$ or $\flat$.
As levap says, this is the adjoint, and you want your vector spaces to be inner product spaces but you haven't specified that. My preferred convention is to use $T^{\ast} : W^{\ast} \to V^{\ast}$ for the dual and $T^{\dagger} : W \to V$ for the adjoint but this isn't universal.