Consider a group $\,G\,$, a vector space $\,{\mathbb{V}}\,$, and a space $\,{\cal{L}}^G\,$ of functions $\varphi$ on this group: $$ {\cal{L}}^G\;=\;\left\{~\varphi~\Big{|}~~~\varphi:\,~G\longrightarrow{\mathbb{V}}\,\right\}~~. $$ Let $\,D\,$ be a representation of a subgroup $\,K\leq G\,$ in the said vector space: $$ D~:\quad K~\longrightarrow~GL({\mathbb{V}})\;\;.\qquad\qquad\qquad \label{1}\tag{1} $$
The induced representation $\,D(K)\uparrow G\,=\,\operatorname{Ind}_K^GD\,$ is implemented with the left translations $$ U_g\varphi(x)=\varphi({g^{-1}}x)~~,\qquad g,\,x\in G\;, $$ acting in the subspace comprising Mackey functions (those satisfying the equivariance condition): $$ \left\{~\varphi~\Big{|}~~~\varphi:\,~G\longrightarrow{\mathbb{V}}\;,\quad \varphi(xk)=D^{-1}(k)\,\varphi(x)\,\right\}~\subset~{\cal{L}}^G~~,~~~x\in G~~,~~~k\in K~~.~~~~~\label{2}\tag{2} $$
QUESTION 1.
For the space of Mackey functions, I saw two notations: $\;\operatorname{Hom}_K(G,\,{\mathbb{V}})\;$ and $\;\operatorname{Map}_K(G,\,{\mathbb{V}})\;$.
What is the difference between these notations?
Are both equally acceptable?
QUESTION 2.
Space \eqref{2} of Mackey functions is a section over the associated vector bundle over the left coset space $G/K$. Consequently, the space of Mackey functions splits into $\,|G/K|\,$ subspaces: $$ \operatorname{Map}_K(G,\,{\mathbb{V}})\,=\;\bigoplus_X \, \operatorname{Map}_K(X,\,{\mathbb{V}}) \;\;,\qquad\qquad $$ where $X\in G/K$ is a left coset, while the fiber above this coset, $$ \operatorname{Map}_K(X,\,{\mathbb{V}})\,=\;\left\{\,\varphi_{_X}\;\Big{|}\;\varphi_{_X}\,=\;P_{_X}\,\varphi\; \right\}\;\;,\;\;\;\; $$ comprises such functions $\varphi_{_X}$ that each $\varphi_{_X}$ coincides with $\varphi$ on a coset $X$ and is zero outside it: $$ \varphi_{_X}\,\equiv\;P_{_X}\,\varphi\;=\; \left\{ \begin{array}{lll} \varphi(g)\;\;\;&\mbox{for}\quad g\in X\;\;,\\ 0\;\;\;\;\;&\mbox{for}\quad g\notin X\;\;. \end{array} \right. $$ Mind that each such $\varphi_{_X}$ is a Mackey function, because $\, D(k)\,\varphi_{_X}(g)=\varphi_{_X}(g\, k^{-1})$ holds for it.
Now, the question:
With a left coset $X$ understood as a point of the base $G/K$, was it correct of me to employ the notation $\;\operatorname{Map}_K(X,\,{\mathbb{V}})\;$ for the fiber corresponding to $X\,$?
Could I have used for this fiber also the notation $\;\operatorname{Hom}_K(X,\,{\mathbb{V}})\;$?