My background is severely lacking in tensor algebra, and after a few days of looking into tensors I am still not able to even formulate this question quite correctly; my apologies for that. I am aware upper and lower indices are used for a specific purpose in tensor calculus but I will here use them differently.
I have a certain product of a three dimensional array (tensor?) with components $\psi^s_{kl} $ with a matrix with components $\phi^s_l$ which works by multiplying the $s$'th matrix of $\Psi$ with the $s$'th vector of $\Phi$: \begin{equation} \sum_l \psi^s_{kl} \phi^s_l \end{equation}
My question is whether this kind of product has a name, and if so what is it and how can I write it down using tensor notation. Any other helpful resources for this kind of operation would be appreciated.