[Tex/LaTex] Fine-tuning a commutative diagram

diagramstikz-pgf

I am trying to type the following simple commutative diagram in LaTeX. I have the basics, but I need for the V to be centered, and for the arrow under the g to be in the opposite direction.

\begin{tikzpicture}[every node/.style={midway}]
\matrix[column sep={4em,between origins},
    row sep={2em}] at (0,0)
{ \node(X)   {$X$}  ; & \node(Y) {$Y$}; \\
\node(V) {$V$};                   \\};
\draw[<-] (V) -- (X) node[anchor=east]  {$h$};
\draw[->] (V) -- (Y) node[anchor=north]  {$g$};
\draw[->] (X)   -- (Y) node[anchor=south] {$f$};
\end{tikzpicture}

Best Answer

Does this solution with tikz-cd help?

\documentclass{standalone}
\usepackage{tikz-cd}

\begin{document}

\begin{tikzcd}[column sep=small]
X \arrow{rr}{f} \arrow[swap]{dr}{h}& &Y \arrow{dl}{g}\\
& V & 
\end{tikzcd}    

\end{document}  

enter image description here