What is the meaning of $(\mu \otimes \mu) ({\rm id}\otimes \tau \otimes {\rm id})$?

Question

Here is one of the figures whose commutativity express the fact that $\Delta$ is a morphism of algebra:

$\require{AMScd}$ $$\begin{CD}H\otimes H @>\Delta\otimes\Delta>> (H\otimes H)\otimes (H\otimes H)\\ @V \mu V V @VV (\mu\otimes\mu)({\rm id}\otimes \tau\otimes{\rm id}) V\\ H @>>\Delta > H\otimes H. \end{CD}$$

(Source: Christian Kassel, Quantum Groups, p.45.)

But I do not quite sure understand the following notation, what is the meaning of $(\mu \otimes \mu) ({\rm id} \otimes \tau \otimes {\rm id}) $? how should I apply it? Could someone explain this to me, please?

Answer 1

$\tau$ here is the switch map $$H\otimes H \to H \otimes H: h \otimes g \mapsto g \otimes h$$ you then form $$\operatorname{id}\otimes \tau \otimes \operatorname{id}: H \otimes H \otimes H \otimes H \to H \otimes H \otimes H \otimes H: h \otimes g \otimes k \otimes l \mapsto h \otimes k \otimes g \otimes l.$$

$\mu$ is the multiplication map of the algebra $$\mu: H \otimes H \to H: h\otimes g \mapsto hg$$ Thus $$\mu\otimes \mu: H \otimes H \otimes H \otimes H \to H \otimes H: g\otimes h \otimes k\otimes l \mapsto gh \otimes kl$$

You then consider the composition $$(\mu\otimes \mu)\circ (\operatorname{id}\otimes \tau \otimes \operatorname{id}): H \otimes H \otimes H \otimes H \to H\otimes H$$ which maps $$g\otimes h \otimes k \otimes l \mapsto gk \otimes hl.$$