# How to derive the way a controlled $U$ gate acts on an input state?

Wikipedia states that a controlled U gate maps the basis states to the following:

$$|00\rangle \mapsto |00\rangle\\ |01\rangle \mapsto |01\rangle\\ |10\rangle \mapsto |1 \rangle \otimes U |0 \rangle = |1 \rangle \otimes \left(u_{00}|0 \rangle + u_{10}|1 \rangle\right)\\ |11\rangle \mapsto |1 \rangle \otimes U |1 \rangle = |1 \rangle \otimes \left(u_{01}|0 \rangle + x_{11} | 1 \rangle\right)$$

I have looked at Nielsen and Chuang - Quantum Computation and Information 10$$^{th}$$; but I wasn't able to find the derivation process.
It is not a derived mapping, it's the mathematical version of the definition of how a controlled gate works: it does nothing to the second qubit if the first qubit is $$|0\rangle$$, and applies the unitary $$U$$ to the second qubit if the first qubit is $$|1\rangle$$.