A smooth map $f\colon M\to B$ between compact manifolds is a fibration if it is surjective and submersive. I would like to use the same definition when $M$ and $B$ have boundary. Write $$ \partial M = \partial_{\mathrm{vert}}M\cup \partial_{\mathrm{hor}}M \quad\mbox{where}\quad \partial_{\mathrm{vert}}M=f^{-1}(\partial B),\quad \partial_{\mathrm{hor}}M=\overline{\partial M\setminus \partial_{\mathrm{vert}}M}. $$ Is it true that $f|_{\partial_{\mathrm{hor}}M}\colon \partial_{\mathrm{hor}}M\to B$ is a fibration, too?
If yes, I would like to ask if it is a standard fact, or there is some reference where it is clearly stated. If no, is there a standard definition of fibrations between manifolds with boundary for which one has a decomposition like above?