Standardize API auth challenge responses as ProblemDetails

Program.cs

@@ -2,6 +2,7 @@ using GameList.Data;
 using GameList.Endpoints;
 using GameList.Infrastructure;
 using Microsoft.AspNetCore.Authentication.Cookies;
+using Microsoft.AspNetCore.Mvc;
 using Microsoft.AspNetCore.DataProtection;
 using Microsoft.AspNetCore.HttpOverrides;
 using Microsoft.Data.Sqlite;
@@ -54,16 +55,8 @@ builder.Services.AddAuthentication(CookieAuthenticationDefaults.AuthenticationSc
     options.ExpireTimeSpan = TimeSpan.FromDays(30);
     options.Events = new CookieAuthenticationEvents
     {
-        OnRedirectToLogin = ctx =>
-        {
-            ctx.Response.StatusCode = StatusCodes.Status401Unauthorized;
-            return Task.CompletedTask;
-        },
-        OnRedirectToAccessDenied = ctx =>
-        {
-            ctx.Response.StatusCode = StatusCodes.Status401Unauthorized;
-            return Task.CompletedTask;
-        }
+        OnRedirectToLogin = ctx => WriteUnauthorizedChallengeAsync(ctx.HttpContext),
+        OnRedirectToAccessDenied = ctx => WriteUnauthorizedChallengeAsync(ctx.HttpContext)
     };
 });
 
@@ -139,6 +132,26 @@ static ForwardedHeadersOptions BuildForwardedHeadersOptions(IConfiguration confi
     return options;
 }
 
+static Task WriteUnauthorizedChallengeAsync(HttpContext context)
+{
+    context.Response.StatusCode = StatusCodes.Status401Unauthorized;
+    if (!context.Request.Path.StartsWithSegments("/api"))
+        return Task.CompletedTask;
+
+    if (context.Response.HasStarted)
+        return Task.CompletedTask;
+
+    context.Response.ContentType = "application/problem+json";
+    var problem = new ProblemDetails
+    {
+        Status = StatusCodes.Status401Unauthorized,
+        Title = "Unauthorized",
+        Detail = "Unauthorized"
+    };
+    problem.Extensions["error"] = "Unauthorized";
+    return context.Response.WriteAsJsonAsync(problem);
+}
+
 static void UpdateIndexMetaBase(IWebHostEnvironment env, string basePath)
 {
     try