Is a canonical projection always continuous in the quotient topological space

Question

I'm solving some exercises on quotient topology, and I wonder if a canonical surjection is always continuous.

Let $\pi: (X,\tau_X)\to (X/_\sim,\tau_\sim)$ be a canonical surjection.

I think $\pi$ is continuous, because of the following.

Let $U\in \tau_\sim$ then by definition of $\tau_\sim := \{ V\in X/_\sim: \pi^{-1}(V)\in \tau_X\}$ the inverse $\pi^{-1}(U)\in \tau_X$ which would imply $\pi$ to be continuous.

Is this true?