How of to change font size for every nodes in tikzcd?

Since the TikZ cd is a TikZ matrix, you could use every matrix/.append style={nodes={font=...}}.

I would also leave out all those \! in the first label.

\documentclass{article} 
\usepackage{amsmath} 
\usepackage{tikz-cd} 
\usetikzlibrary{arrows, matrix} 
\tikzcdset{every label/.append style = {scale=0.6,yshift=0.6ex},
every matrix/.append style={nodes={font=\small}}
} 
\begin{document} 
\[\begin{tikzcd} 0\rar & \mathrm{Hom}_R(L, E) \arrow{r}{\mathrm{Hom}(\beta, \mathrm{id}_E)} & \mathrm{Hom}_R(N, E) \arrow{r}{\mathrm{Hom}(\alpha, \mathrm{id}_E)} & \mathrm{Hom}_R(M, E) \end{tikzcd}\] 
\end{document}

Edit: As marmot pointed out, there is also the key cells=<options> which appends <options> to the style every cell, which, in turn, by default contains the following:

/tikz/shape=asymmetrical rectangle,
/tikz/inner xsep=1ex,
/tikz/inner ysep=0.85ex

See the tikzcd package documentation, Section 2.1 General options.

MWE:

\documentclass{article} 
\usepackage{amsmath} 
\usepackage{tikz-cd} 
\usetikzlibrary{arrows, matrix} 
\tikzcdset{
    every label/.append style = {scale=0.6,yshift=0.6ex},
    cells={nodes={font=\small}}
    } 
\begin{document} 
\[
\begin{tikzcd} 
0\rar & \mathrm{Hom}_R(L, E) \arrow{r}{\mathrm{Hom}(\beta, \mathrm{id}_E)} & \mathrm{Hom}_R(N, E) \arrow{r}{\mathrm{Hom}(\alpha, \mathrm{id}_E)} & \mathrm{Hom}_R(M, E) 
\end{tikzcd}
\] 
\end{document}

The result is the same as before.

\documentclass{article} 
\usepackage{amsmath} 
\usepackage{tikz-cd} 
\usetikzlibrary{arrows, matrix} 
\tikzcdset{every label/.append style = {scale=0.6,yshift=0.2ex},
    nodes={font=\footnotesize}} 
\begin{document} 
\[\begin{tikzcd} 0\rar & \mathrm{Hom}_R(L, E) \arrow{r}{\mathrm{Hom}\!(\!\beta\!, \mathrm{id}\!_E\!)} & \mathrm{Hom}_R(N, E) \arrow{r}{\mathrm{Hom}(\alpha, \mathrm{id}_E)} & \mathrm{Hom}_R(M, E) \end{tikzcd}\] 
\end{document}

Btw how can readers read your supertiny text? I can't read it.