What role does the nylon patch attached to the elbow play?

Looking first at where the patch is placed reveals the answer. The nylon patch of Symmetric E-patch Sweat #7 7 is placed on the elbow, and its purpose is to minimize fraying in that area^[1].

The elbow is an area where friction concentrates at a single point, as the fabric stretches every time you bend your arm, combined with actions like leaning on a desk or armrest. Cotton fleece fabric, commonly used in sweatshirts, feels soft, but its surface fibers tend to fluff up and pill first, and if that condition continues, only that specific spot becomes thin. Nylon has relatively high tensile strength and abrasion resistance among synthetic fibers, so it has long been used as reinforcement in places subject to frequent rubbing, such as the bottom of backpacks or the knees of workwear.

In other words, the elbow patch is a method of placing a layer of strong material over a weak area. However, nylon is vulnerable to heat and easily damaged by high-temperature ironing or dryer use, so it is safer to avoid that part when caring for it.

hmmm, this kind of reinforcement is the type of detail whose difference shows when looking at the sleeves again after several seasons have passed, rather than when wearing the garment for the first time.