Trace of a morphism of sheaves

$ \underline {Background}$ :Suppose $ X$ be a scheme and $ \mathcal F$ and $ \mathcal G$ are two sheaves of $ \mathcal O_X$ modules.

Also assume that $ \exists $ {U}$ $ a cover of $ X$ by open sets such that each $ \mathcal F(U),\mathcal G(U)$ are finitely free $ \mathcal O(U)$ modules.

Let $ \exists \phi:\mathcal F\to\mathcal G$ a sheaf of $ \mathcal O_X$ modules.

$ \underline {Question}$ :what is the meaning of the statement “trace$ \phi=0$

$ \underline {Guess}$ :Does it mean for all open set $ V$ of $ X$

$ \phi(V):\mathcal F(V)\to \mathcal G(V)$ has trace of the matrix of $ \phi(V)=0$

But this makes sense only for $ V=U$ because,we only have $ \phi(U)$ is a morphism between 2 finitely free $ \mathcal O(U)$ modules,and hence its matrix w.r.to canonical bases exists.

So,I would like to know the appropriate definition of trace being $ 0$ and any text which mentions it clearly.

Any help from anyone is welcome.