The coprocessor of the 80386 microprocessor is the 80387 coprocessor.

The 80386 microprocessor is a 32-bit microprocessor introduced by Intel in the mid-1980s. It was a significant improvement over its predecessor, the 80286, and offered enhanced performance and capabilities. One of the key features of the 80386 was its ability to support a coprocessor.

A coprocessor is a specialized processor that works alongside the main processor (in this case, the 80386) to perform specific tasks. It is designed to offload certain types of computations, such as floating-point arithmetic, from the main processor, thereby improving overall system performance.

The 80387 coprocessor, also known as the 387 or the math coprocessor, is specifically designed to handle floating-point arithmetic operations. Floating-point arithmetic involves performing calculations with numbers that have fractional parts or very large or very small values. Examples of such calculations include scientific computations, graphics rendering, and complex mathematical calculations.

The 80387 coprocessor is compatible with the 80386 microprocessor and can be used in conjunction with it to improve the performance of floating-point calculations. It provides additional instructions and registers specifically tailored for floating-point arithmetic operations.

The other options listed in the question, namely 8087, 8089, and 8088, are not coprocessors of the 80386 microprocessor.

- The 8087 is a coprocessor designed for the 8086 and 8088 microprocessors, which are predecessors of the 80386.
- The 8089 is a direct memory access (DMA) controller, not a coprocessor.
- The 8088 is a microprocessor that is similar to the 8086 but with an 8-bit external data bus.

Therefore, the correct answer is option A, the 80387 coprocessor, as it is the coprocessor specifically designed for the 80386 microprocessor.