How can q(z,l,u,c|x) be factorized to q(z|x)q(l|x)q(c|z)q(u|c,z)? How can q(z,l,u,c|x) be factorized to q(z|x)q(l|x)q(c|z)q(u|c,z)?
I don't know what assumptions are behind this equation, which is I want to know. I think normal chain rule of probability is in the process, but only chain rule can't derive the formula.
What I know is that z, l, u follows gaussian distribution, and c follows multinomial distribution.
It's from the paper Xu C, Lopez R, Mehlman E, Regier J, Jordan MI, Yosef N. Probabilistic harmonization and annotation of single-cell transcriptomics data with deep generative models. Mol Syst Biol. 2021 Jan;17(1):e9620. doi: 10.15252/msb.20209620. PMID: 33491336; PMCID: PMC7829634.
That formula is used for derive ELBO, since the model is similar to VAE.

It can't be factorised that way.  The paper doesn't claim it can. What they say is that they are computing a variational approximation using a product of that form. It's fairly common for variational approximations to be products, even when the true posterior or likelihood doesn't factorise as a product of independent components.
