Suppose that for $\theta \in R$ we have $f(z)=g(z,\theta)$. Now fix $z=z_0$, is $g(z_0,\theta)=f(z_0)$ for all $\theta$ in its domain?

Question

Let $f(z)$ and $g(z,\theta)$ be complex functions with $f$ analytic and $g$ meromorphic. Suppose that for $\theta \in R$ we have $f(z)=g(z,\theta)$. Now fix $z=z_0$, is $g(z_0,\theta)=f(z_0)$ for all $\theta$ in its domain ?

I am supposing that they are equal by the identity theorem of meromorphic function but I am not sure