This will largely depend on how each serum was formulated and which form of vitamin C is in your serum.
Vitamin C, in its active form, ascorbic acid, is acidic (unsurprisingly!). This means that it needs to be mixed with other ingredients that are also happy in an acidic environment. If you look on the ingredients list of the vitamin C serum, it will say whether it is ascorbic acid or another form. The most often used forms are:
L-Ascorbic Acid: which is effective at a lower pH, typically around 2.0 to 3.5.
Magnesium Ascorbyl Phosphate (MAP): MAP is more stable and less pH-sensitive. It works well at pH range of around 5.5 to 7.0, which is close to the skin's natural pH and so less irritating.
Sodium Ascorbyl Phosphate (SAP): Similar to MAP.
Ascorbyl Palmitate: is pretty stable at around 6.0 to 7.0.
Tetrahexyldecyl Ascorbate: This one needs to be in an oil base, so if the serum feels very oily, this one may have been used. It is comfortable at a pH of 5.0 to 7.0, similar to ascorbyl palmitate. (An oily formula, without water, isn’t described in terms of pH.
If it isn’t L’ascorbic, then I can’t see pH being a problem.
These two ingredients can and are formulated together in a serum or cream, so there’s nothing about them that means you shouldn’t combine them. But each will have been made so that the ingredients in the product with them work to the best effect, so you may not need both.
The only problem I can see, in using one and then the other immediately, is that both vitamins can be drying and irritating over a certain dilution. I would expect that if they are in a high percentage, there are other ingredients in there to counter drying or irritation. If not, invest in a good moisturiser, with hyaluronic acid, sodium PCA, aloe vera or another inexpensive humectant, with some oils and if you don’t suffer from oily skin or acne, some butters too and that should sooth any dryness the two vitamins could cause.
(If you are using retinol, I expect it was prescribed, so just ask whoever prescribed it whether your skin also needs a vitamin C serum.)