I'm trying to optimize a function $F\circ A$ where $F\colon \mathbb{R}^{n}\to\mathbb{R}$ and $A\in GL_{n}(\mathbb{R})$. I take a third derivative and and I get $H=D^{3}F_{(A(x))} \circ A^{3}$. Here, $A^{3}\in L(\mathbb{R}^{n},\mathbb{R}^{n})$ and if $T:=D^{3}F_{A(x)}$ then $T\in L(\mathbb{R}^{n},L(\mathbb{R^{n}},L(\mathbb{R}^{n},\mathbb{R})))$ so $H\in L(\mathbb{R}^{n},L(\mathbb{R^{n}},L(\mathbb{R}^{n},\mathbb{R})))$.
My question is how do I actually compose the composition of $T$ and $A^{3}$ as a multidimensional array of numbers (of type $n\times n\times n$). If this was the second derivative, this would be matrix multiplication. What is the analog for a tensor or vector of matrices and a matrix?