I am wondering the interpretation of $|\operatorname{coker}\phi|$, where $$\phi:\mathbb{Z}[t]^n\rightarrow\mathbb{Z}[t]^n,(p_1(t),...,p_n(t))\mapsto(p_1(t),...,p_n(t))\cdot A,$$ and $A$ is an $n\times n$ matrix over $\mathbb{Z}[t]$. Namely, $\phi$ is a linear map of $\mathbb{Z}[t]$-modules.
I know in the case of $\psi:\mathbb{Z}^n\rightarrow\mathbb{Z}^n,$ given that $\det \psi\neq 0$, we have $|\det\psi|=|\operatorname{coker}\psi|$. It has a clear geometric interpretation.
However, in the case of polynomial rings, we could have infinite $|\operatorname{coker}\phi|$ with non-vanished $\det A$. For example, $A=tI$.
My question: in this case, how should we interpret $|\operatorname{coker}\phi|$ and how could we relate it with $\det A$?