It is known that, if \(A\) is an \(n \times n\) complex matrix with no eigenvalues on the closed negative real axis and \(B\) is an arbitrary square complex matrix of order \(n\), then the matrix-matrix exponentiation \(A^B\) is defined as \[ A^B=e^{\log{(A)}B}, \] where \(e^X\) stands for the exponential of the matrix \(X\) and \(\log{(A)}\) denotes the principal logarithm of \(A\). In this paper, \(A^B\) is regarded as a function from \(\mathbb{C}^{n \times n} \times \mathbb{C}^{n \times n}\) to \(\mathbb{C}^{n \times n}\) which assigns to each pair of matrices \((A,B)\) the \(n \times n\) complex matrix \(A^B\). Some facts about the matrix-matrix exponentiation are revised and new results are presented. The authors introduce a new concept of bivariate matrix functions, which is illustrated with some examples, among others, the matrix-matrix exponentiation. Formulae for Fréchet derivatives of bivariate matrix functions and conditions for their existence are given in this manuscript. Those formulae are used in turn to derive a closed representation of the Fréchet derivative of the matrix-matrix exponentiation. A section of this paper is devoted to analyze the matrix-matrix exponentiation conditioning, where the key is the norm of operator \(L_f(A,B)\) which denotes the Fréchet derivative of map \(f: \mathbb{C}^{n \times n} \times \mathbb{C}^{n \times n} \rightarrow \mathbb{C}^{n \times n}\) at \((A,B)\). In particular, an algorithm for estimating the related condition number is proposed. The performance of this algorithm is illustrated by several numerical experiments that confirm the theoretical results.