Proof of $[[M'\to M\to M''\to 0]]\Rarr[[M'\otimes N\to M\otimes N\to M''\otimes N\to 0]]$

$\ast$ definition of cokernels:
$$ \big[M'\to M\to M''\to 0\big] \text{ exact} \newline \Updownarrow\newline \forall P: \big((M'\to M\xrightarrow{f} N)=0 \big) \Rightarrow \big(f=(M\to M''\to N) \big)\newline \Updownarrow\newline \big[0\to\mathrm{Hom}(M'',N)\to\mathrm{Hom}(M,N)\to \mathrm{Hom}(M',N)\big] \text{ exact} $$ $\ast$ bilinear property: for given $f:M\otimes N\to P$
    fix $x\in M$, then $f(x,\ast):y\mapsto f(x,y)\in\mathrm{Hom}(N,P)$
$\Large\leadsto$ $$ %\begin{align*} M'\to M\to M''\to 0 \text{ exact}\\ \big\Downarrow\\ 0\to\mathrm{Hom}(M',\mathrm{Hom}(N,P))\to\mathrm{Hom}(M,\mathrm{Hom}(N,P))\to\mathrm{Hom}(M'',\mathrm{Hom}(N,P))\\ \text{exact},\,\forall N,P\\ \big\updownarrow\\ 0\to\mathrm{Hom}(M'\otimes N,P)\to\mathrm{Hom}(M\otimes N,P)\to\mathrm{Hom}(M''\otimes N,P)\\ \text{exact},\,\forall P\\ \big\updownarrow\\ M'\otimes N\to M\otimes N\to M''\otimes N\to 0 \text{ exact} %\end{align*} $$

Direct sum $\sum_{i\in I}M_i$ and direct product $\prod_{i\in I}M_i$ are therefore the same if the index set $I$ is finite, but not otherwise, in general.
[Chapter 2 Modules :: direct sum and direct product]

You can also reach the same point if you define direct sum and direct product as direct limit $\varinjlim$ and inverse limit $\varprojlim$ (i.e.)