Continuous map from topological space $A\rightarrow B$ as composition of quotient map and unique continuous map from quotient space to $B$

Question

Let $(A,T_A)$ be a topological space; $R$ an equivalence relation; $Q=A/R$ the quotient set with $i:A\rightarrow Q$ the quotient map and $T_Q=\{V\subset Q|i^{-1}(V)\text{ open in $X$}\}$ the quotient topology.
Let $B$ be a topological space with $\phi:A\rightarrow B$ a continuous map such that for all $a,a'\in A$ we have $aRa'\implies\phi(a)=\phi(a')$.

How do I prove that there exists exactly one continuous map $\psi:Q\rightarrow B$ with $\phi=\psi\circ i$?

What I know:
I know that if $aRa'$ then we get $i(a)=i(a')$. Would it be enough to define $\psi(i(a)):=\phi(a)$ and prove that it is continuous?
And how would I prove continuity and uniqueness?