Inclusion arrow in TikZ

You can use the right hook-latex , or right hook-> arrow (depending on the arrow type you want on the other side) from the arrows library:

Code:

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{arrows}
\begin{document}
\begin{tikzpicture}
  \draw [ultra thick, right hook-latex, red] (0, 0) -- (3, 0);     
  \draw [ultra thick, right hook->,    blue] (0,-1) -- (3,-1);
\end{tikzpicture}
\end{document}

For commutative diagrams I suggest using the tikz-cd package; it uses TikZ to facilitate the drawing of commutative diagrams (it has its own arrows library designed for diagrams and the arrow you are looking for is already buil-in); a little example with the requested inclusion (and some others just for illustration):

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

\begin{document}

\begin{tikzcd}
0 \arrow[hookrightarrow]{r} 
  & A\arrow[hookrightarrow]{r}{\varphi}\arrow[hookrightarrow]{d}{\eta_1} 
  & B\arrow{r}{\psi}\arrow{d}{\eta_2} 
  & C\arrow{r}\arrow{d}{\eta_3} 
  & 0 \\
0 \arrow[hookrightarrow]{r} 
  & A'\arrow[hookrightarrow]{r}[swap]{\varphi'} 
  & B'\arrow{r}[swap]{\psi'} 
  & C'\arrow{r}
  & 0 
\end{tikzcd}
\end{document}

You can use the arrows library from TikZ, so for example, to use the -latex style from TikZ, you can say

\documentclass{article}
\usepackage{tikz-cd}
\usetikzlibrary{arrows}

\tikzset{
commutative diagrams/.cd,
arrow style=tikz,
diagrams={>=latex}}

\begin{document}


\begin{tikzcd}
0 \arrow[hookrightarrow]{r} 
  & A\arrow[hookrightarrow]{r}{\varphi}\arrow[hookrightarrow]{d}{\eta_1} 
  & B\arrow{r}{\psi}\arrow{d}{\eta_2} 
  & C\arrow{r}\arrow{d}{\eta_3} 
  & 0 \\
0 \arrow[hookrightarrow]{r} 
  & A'\arrow[hookrightarrow]{r}[swap]{\varphi'} 
  & B'\arrow{r}[swap]{\psi'} 
  & C'\arrow{r}
  & 0 
\end{tikzcd}
\end{document}

The above code changes the tip arrow style for all the diagrams, but you can select the -latex arrow tip just for some arrows (perhaps not really desirable to have two different arrow tips on the same diagram):

\arrow[hookrightarrow,-latex]{d}{\eta_1}