In Crosby v. Buchanan, the United States Supreme Court was asked to decide whether a state court had the authority to issue a writ of habeas corpus to a prisoner who had been convicted in a federal court. The petitioner, Crosby, had been convicted in a federal court of a crime and was serving his sentence in a state prison. He then filed a petition for a writ of habeas corpus in the state court, arguing that his conviction was unconstitutional. The state court granted the writ and ordered Crosby released from prison. The Supreme Court held that the state court did not have the authority to issue the writ of habeas corpus. The Court reasoned that the writ of habeas corpus is a federal remedy and that the state court did not have the power to interfere with a federal conviction. The Court further held that the state court's decision was not binding on the federal court and that the federal court was the only court that could decide the constitutionality of the conviction. In conclusion, the Supreme Court held that the state court did not have the authority to issue a writ of habeas corpus to a prisoner who had been convicted in a federal court. The Court reasoned that the writ of habeas corpus is a federal remedy and that the state court did not have the power to interfere with a federal conviction. The Court further held that the state court's decision was not binding on the federal court and that the federal court was the only court that could decide the constitutionality of the conviction.

In the case of Crosby v. Buchanan, the Supreme Court was tasked with determining whether a state court had jurisdiction to hear a suit brought by an alien against another citizen of that same state. The majority opinion held that it did not have such jurisdiction and dismissed the appeal. Justice Field dissented from this decision, arguing that there is no constitutional provision or statute which prohibits aliens from bringing suits in state courts against citizens of those states. He argued further that if Congress has not seen fit to pass any laws prohibiting such actions then it should be left up to each individual state's legislature as they are best suited for making decisions regarding their own judicial systems and procedures. Furthermore, he noted how denying aliens access to justice would be contrary to our nation’s commitment towards providing equal protection under law regardless of one’s citizenship status or national origin. Therefore, Justice Field concluded his dissent by stating his belief that allowing aliens access into our nation’s courts is essential for upholding fundamental principles upon which our country was founded on – namely liberty and justice for all individuals within its borders